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For to say that something ϕ’s in the domain as enlarged by α is to say that something suitably related to α is a ϕ; and I cannot make proper sense of what this ‘something’ might be unless I have already enlarged the domain by α. ¹⁹ This feature of the postulationism might be thought to be at odds with our previous insistence that a postulate should serve to reinterpret the quantifier. For surely, if I reinterpret the quantifier, then what I say, before laying down a postulate, is different from what I say afterwards.

A somewhat similar line of argument is given by Dummett (1991), pp. 315–16. 26 Kit Fine Let us begin by reformulating the original argument, making reference to the interpretations explicit. Presumably, our opponent’s intended use of the quantifier will conform to a particular interpretation I0 of the quantifier. We may therefore assume: (1) ∀x∃I0 y(y = x)&∀I0 y∃x(x = y). We now produce an ‘extension’ J0 of I0 subject to the following condition: (2) ∃J0 y∀I0 x(x ∈ y ≡ ∼x ∈ x). From (2) we may derive: (3) ∃J0 y∀I0 x(x = y).

Or again, it has been suggested that we should think of sets as being constructed in stages and that what prevents the formation of the universal or the Russell set is there being no stage at which its members are all constructed. We may grant that we should think of sets as being constructed at stages and that, under any reasonable process by which we might take them to be constructed, there will be no stage at which either the universal or the Russell set is constructed. But what is to prevent us from so understanding the quantifier over stages that it includes a stage that lies after all of the stages according to the original understanding of the quantifier (∃+ α∀β(α > β) )?